The autornorphisrn group of smcoth G-manifokls andals applications

设M是一个光滑流形，它具有紧李群G的光滑作用。设Lip_G（M）表示M的等变Lipschitz同胚群.在Lip_G（M）上可以引入两种拓扑.设Lip_G（M），（resp. L_G（M））表示当Lip_G（M）具有紧开Lipschitz拓扑时Lip_G（M）的单位元的单位分支.紧凑开放拓扑）。研究了H_I（Lip_G（M）_O）的第一同调群，当M是主G-流形或G是有限群时，证明了群LiP_G（M）_O是完全群.当M是具有标准U（n）作用的复n维空间C^n时，H_1（LG（M））同构于某些具有连续模的真实的值函数的向量空间.首先证明了当V是余维为1轨道的表示空间时，Lip_G（V）_o是完美的。利用这一结果并分析了等变Lipschitz同胚在奇异轨道周围的行为，我们计算了任意余维为1轨道的光滑G流形的H_1（Lip_G（M）-O）。结果表明，光滑G流形的自同构群的第一同调群与光滑、紧开或紧开Lipschitz等范畴有很大的依赖关系。我们还计算了三维光滑S^I流形的等变同同态群的第一同调群。这一成果最近发表在国外期刊上。

Let M be a smooth manifold which has a smooth action of a compact Lie group G. Let Lip_G(M) denote the group of equivariant Lipschitz homeomorphisms of M. We can introduce two kind of topologies on Lip_G(M). Let Lip_G(M), (resp. L_G(M)) denote the identity component of the identity of Lip_G(M) when Lip_G(M) has the compact open Lipschitz topology (resp. compact open topology). We investigate the first homology group of H_I(Lip_G(M)_O).When M is a principal G-manifold or G is a finite group, it is known that the group LiP_G(M)_O is perfect. When M is the complex n-dimensional space C^n with the canonical U(n)-action,H_1(L_G(M)) is isomorphic to the vector space of some real valued functions which has continuous moduli.During the study duration, we investigate the case where M has a G-manifold with codimension one orbit. First we proved that Lip_G(V)_o is perfect when V is a representation space with codimension one orbit. By using this result and analyzing the behavior of equivariant Lipschitz homeomorphism around the singular orbit, we calculate H_1(Lip_G(M)-O) for any smooth Gmanifold with codimension one orbit. The result shows that the first homology group of the automorphism group of smooth Gmanifold is quite depends on the category such as smooth, compact open or compact open Lipschitz. We also calculate the first homology group of the group of equivariant diffeomorphisms of 3-dimensional smooth S^Imanifold. The result has recently published in the foreign journal.